The profit function \( P(x) = 20x - 6000 \) represents the profit from the production and sale of specialty golf hats, where \( x \) is the number of hats produced and sold.
Key Components of the Function:
- Coefficient of \( x \): The term \( 20x \) indicates that for each hat sold, the profit increases by $20.
- Constant Term: The constant \( -6000 \) represents the fixed costs associated with production, which must be covered before any profit can be realized.
Finding the Break-Even Point:
To determine the break-even point, where profit \( P(x) = 0 \):
\[ 20x - 6000 = 0 \]
Solving for \( x \):
\[ 20x = 6000 \] \[ x = \frac{6000}{20} = 300 \]
This means that 300 hats must be produced and sold to cover all costs, and any sales beyond this point will contribute to profit.
Profit Calculation:
- For different values of \( x \):
- If \( x = 0 \) (no hats sold), \( P(0) = 20(0) - 6000 = -6000 \) (loss of $6000).
- If \( x = 300 \) (break-even), \( P(300) = 20(300) - 6000 = 0 \) (no profit, no loss).
- If \( x = 400 \) (selling 400 hats), \( P(400) = 20(400) - 6000 = 8000 \) (profit of $8000).
Summary:
- The profit increases linearly with each additional hat sold.
- The break-even point is at \( x = 300 \).
- The profit function indicates that below 300 hats sold results in a loss, while selling more than 300 hats results in profit.